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In this paper, we prove the degenerations of Schubert varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to (normal) toric varieties.


Well known wonderfulGvarieties are those of rank zero, namely the generalized flag varietiesG/P, those of rank one, classified in [A], and certain complete symmetric varieties described in [DP] such as the famous space of complete conics.


In this paper we compute the cohomology with trivial coefficients for the Lie superalgebraspsl(n, n), p (n) andq(2n); we show that the cohomology ring ofq(2n+1) is of Krull dimension 1 and we calculate the ring forq(3) andq(5).


As a corollary we obtain af·g·p·d·f subgroup of SLn(?) (n ≧ 3.


More generally, we prove that if Γ is an irreducible arithmetic noncocompact lattice in a higher rank group, then Γ containsf·g·p·d·f groups.


In the last section we give an exposition of results, communicated to us by J.P.


We prove that the moduli space of mathematical instanton bundles on P3 with c2 = 5 is smooth.


We compute the ring of ${\mbox{\rm SL}}(2,{\mbox{\bf R}})$invariants in the ring of polynomial functions, ${\mathcal P}$, on ${\mathcal A}$.


We show that the absolute invariants (i.e.,the ${\mbox{\rm GL}}(2, {\mbox{\bf R}})$invariants in the field of fractions of ${\mathcal P}$) distinguish the isomorphism classes of 2dimensional nonassociative real division algebras.


Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0.


In case p >amp;gt; 0, assume G is defined and split over the finite field of p elements Fp.


Let q be a power of p and let G(q) be the finite group of Fqrational points of G.


Assume B is Fstable, so that U is also Fstable and U(q) is a Sylow psubgroup of G(q).


It is proved that for any prime $p\geqslant 5$ the group $G_2(p)$ is a quotient of $(2,3,7;2p) = \langle X,Y: X^2=Y^3=(XY)^7 =[X,Y]^{2p}=1 \rangle.$


Given integers n,d,e with $1 \leqslant e >amp;lt; \frac{d}{2},$ let $X \subseteq {\Bbb P}^{\binom{d+n}{d}1}$ denote the locus of degree d hypersurfaces in ${\Bbb P}^n$ which are supported on two hyperplanes with multiplicities de and e.


For a finitedimensional representation $\rho: G \rightarrow \mathrm{GL}(M)$ of a group G, the diagonal action of G on $M^p,$ ptuples of elements of M, is usually poorly understood.


Let k be an algebraically closed field of characteristic p ≥ 0.


This result is not true when char k = p >amp;gt; 0 even in the case where H is a torus.


However, we show that the algebra of invariants is always the proot closure of the algebra of polarized invariants.


Let p be a prime and let V be a finitedimensional vector space over the field $\mathbb{F}_p$.




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